Finiteness of elasticity in the one-fixed-point submonoid

Determine whether the multiplicative submonoid of weighted dual skeletons with exactly one fixed point has finite elasticity.

Background

The submonoid consisting of skeletons with one fixed point is divisor-closed and is not half-factorial: the paper constructs an element with factorizations of lengths 2 and 3. In contrast with the full monoid, the infinite-elasticity construction based on forgetting fixed-point counts does not apply within this submonoid, because every factor also has exactly one fixed point.

The paper leaves unresolved whether the elasticity of this restricted submonoid is nevertheless bounded by a finite constant.